pre-calculus lesson 4: the wondrous world of imaginary and complex numbers definitions, operations,...
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Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex NumbersDefinitions, operations, and calculator shortcuts
Definition
The lowercase letter “i” is equal to the square root of negative one.
1i
Examples
9 64216x
Examples
9 64216x
i3
Examples
9 64216x
i3 i8
Examples
9 64216x
i3 i8 xi4
Powers of i
i2 = -1
i3 = -i
i4 = 1
http://supermanjaviolivares.iespana.es/poster1.jpg
Powers of i
i2 = -1
i3 = -i
i4 = 1
Higher powers of i?
http://supermanjaviolivares.iespana.es/poster1.jpg
Let’s exploit 74i
14 i
Let’s exploit 74i 2184 ii
14 i
Let’s exploit 74i 2184 ii
2181 i
14 i
Let’s exploit 74i 2184 ii
2181 i21 i
14 i
Let’s exploit 74i 2184 ii
2181 i21 i
2i
14 i
Let’s exploit 74i 2184 ii
2181 i21 i
2i 1
14 i
Complex NumbersWhat is a complex number?
A number with a real and an imaginary component
Standard form:a + bi
Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5
Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
i93
Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
i93 i39
FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265
FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii
FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii 18310 i
FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii 18310 ii328
Conjugates of Complex Numbers
The conjugate of a+bi is a-bi.
The conjugate of a-bi is a+bi.
Dividing Complex NumbersWe cannot leave i in the
denominator of a fraction. (Why not?)
Multiply the numerator and denominator of the fraction by the conjugate of the denominator.
Sample Problem
i
i
32
65
Sample Problem i
i
32
65
ii
32
32
Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
13
278 i
Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
13
278 i i
13
27
13
8
Graphing Calculator
Press MODE.Select a+bi mode.
The i key is the decimal key.
Sample Problem
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator.
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER.
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac.
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac. 17/10-11/10i
i
i
3
54
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac. 17/10-11/10i
i
i
3
54
IMPORTANT NOTE:
You must WRITE the answer as
i
10
11
10
17
Practice Questions1. (3 + 2i) + (4 – 5i)2. (5 – 6i) – (3 – 2i)3. (2 + 4i)(3 – 5i)4. (4 + i)2 5. 4 + 3i
2 – 3i